By facilitating the generation of samples from arbitrary probability distributions, Markov Chain Monte Carlo (MCMC) is, arguably, \emph{the} tool for the evaluation of Bayesian inference problems that yield non-standard posterior distributions. In recent years, however, it has become apparent that Sequential Monte Carlo (SMC) samplers have the potential to outperform MCMC in a number of ways. SMC samplers are better suited to highly parallel computing architectures and also feature various tuning parameters that are not available to MCMC. One such parameter - the `L-kernel' - is a user-defined probability distribution that can be used to influence the efficiency of the sampler. In the current paper, the authors explain how to derive an expression for the L-kernel that minimises the variance of the estimates realised by an SMC sampler. Various approximation methods are then proposed to aid implementation of the proposed L-kernel. The improved performance of the resulting algorithm is demonstrated in multiple scenarios. For the examples shown in the current paper, the use of an approximately optimum L-kernel has reduced the variance of the SMC estimates by up to 99 % while also reducing the number of times that resampling was required by between 65 % and 70 %. Python code and code tests accompanying this manuscript are available through the Github repository \url{https://github.com/plgreenLIRU/SMC_approx_optL}.
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